I am trying to solve this problem by absurd. I would like to prove that if the circle is placed in any position and it intersects any polygon, the sum of polygons' area should be greater than 100pi. if the circle is placed in any position and it intersect any polygon, this means tha the max interstitial area is designed in such a way that at max a circle of radius 1 can pass though. but how to make a useful evaluation of this area????? as the polygons are convex each polygon can be surrounded by a circular area equal to 2p*2+4pi < 8pi. but 800pi is greater than 38^2..... has someone got a hint?

A classical (today) method. For each polygon, build on each side, towards the exterior of the polygon, a rectangle having one side that side of the polygon, and the other side of length $1$. Then connect these rectangles by circle sectors of radius $1$, around each vertex of the polygon. Thus each polygon is surrounded by a "border" made of all points at a distance of at most $1$ from the polygon. The area of this figure is the area of the polygon (less than $\pi$), plus the total area of the rectangles, clearly equal to the perimeter of the polygon (less than $2\pi$), plus the total area of the circle sectors - but these add up to exactly one full disk of radius $1$ (thus of area $\pi$), since at each vertex the angle of the circle sector is $\pi$ minus the angle at the vertex (so if the polygon has $n$ vertices, the sum of these circle sectors angles is $n\pi - (n-2)\pi = 2\pi$, as the sum of the angles of the polygon is $(n-2)\pi$). Therefore the total area of the $100$ bordered polygons is less than $100(\pi + 2\pi + \pi) = 400\pi < 1257$. Consider now the square of side $36$ obtained from the initial square by cutting off strips of width $1$ along each side; its area is $36^2 = 1296$. Thus it cannot be fully covered by the bordered polygons. A disk of radius $1$ centred at a point not covered will be contained in the initial square, and will not intersect any of the initial polygons (since the distance from its centre to any of the polygons is larger than $1$).

oh yes! it is enough to border the polygons with distance =1 and not 2, as I did.... thanks a lot! rgds ad